On The Infinitude of the Twin Primes

Abstract

We present a novel approach to the Twin Prime Conjecture, basing on the $6x\pm 1$ representation of primes. By defining so-called twin prime generators $x\in \mathbb{N}$, for which both $6x-1$ and $6x+1$ are prime, we reformulate the conjecture into the existence problem of such $x$. Using admissible residue classes modulo products of small primes and an adapted Selberg sieve, we partition the natural numbers into structured intervals $\mc{A}_n$, where the maximal possible prime divisor of $6x\pm 1$ is fixed. Within each $\mc{A}_n$, we apply the sieve to estimate the number of generator candidates that escape all local obstructions. Due to the \emph{parity problem} we cannot solve the problem with a Selberg sieve. It requires other sieves or methods. The author is searching for them and invites all interested people to help.

5 pages